Convolve {2, 1, 5, 7} and {1, 3, 5, 9, 6}

Chosen origin: first sample at $n = 0$ for both: $h[n] = \{\underset{\uparrow}{2}, 1, 5, 7\}$ ($0 \le n \le 3$) and $x[n] = \{\underset{\uparrow}{1}, 3, 5, 9, 6\}$ ($0 \le n \le 4$).

$$y[n] = x[n]*h[n] = \sum_k x[k]\,h[n-k]$$

Output starts at $n = 0$; length $5 + 4 - 1 = 8$, so $0 \le n \le 7$.

Tabular method

Direct working

• $y[0] = 1(2) = 2$
• $y[1] = 1(1) + 3(2) = 7$
• $y[2] = 1(5) + 3(1) + 5(2) = 18$
• $y[3] = 1(7) + 3(5) + 5(1) + 9(2) = 45$
• $y[4] = 3(7) + 5(5) + 9(1) + 6(2) = 67$
• $y[5] = 5(7) + 9(5) + 6(1) = 86$
• $y[6] = 9(7) + 6(5) = 93$
• $y[7] = 6(7) = 42$

Sum check: $(\sum x)(\sum h) = 24 \times 15 = 360$, and $2 + 7 + 18 + 45 + 67 + 86 + 93 + 42 = 360$.

Answer:

$$y[n] = \{\underset{\uparrow}{2},\ 7,\ 18,\ 45,\ 67,\ 86,\ 93,\ 42\}$$